Worksheet: Rotation and Reflection Matrices

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In this worksheet, we will practice finding the matrix of linear transformation which rotates every vector by a given angle and reflects them in the x- or y-axis.

Q1:

Find the matrix with respect to the standard basis vectors for the linear transformation that rotates every vector in ℝ 2 through an angle of 𝜋 3 and then reflects it across the 𝑦 -axis.

  • A ⎛ ⎜ ⎜ ⎜ ⎝ 1 2 √ 3 2 √ 3 2 1 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • B ⎛ ⎜ ⎜ ⎜ ⎝ − √ 3 2 1 2 √ 3 2 1 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • C ⎛ ⎜ ⎜ ⎜ ⎝ 1 2 − √ 3 2 √ 3 2 1 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • D ⎛ ⎜ ⎜ ⎜ ⎝ − 1 2 √ 3 2 √ 3 2 1 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • E ⎛ ⎜ ⎜ ⎜ ⎝ − √ 3 2 1 2 1 2 √ 3 2 ⎞ ⎟ ⎟ ⎟ ⎠

Q2:

A linear transformation is formed by reflecting every vector in ℝ 2 in the 𝑦 -axis and then rotating the resulting vector through an angle of 𝜋 4 . Find the matrix of this linear transformation.

  • A ⎛ ⎜ ⎜ ⎜ ⎝ − √ 2 2 √ 2 2 √ 2 2 √ 2 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • B ⎛ ⎜ ⎜ ⎜ ⎝ − √ 2 2 √ 2 2 − √ 2 2 √ 2 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • C ⎛ ⎜ ⎜ ⎜ ⎝ √ 2 2 − √ 2 2 √ 2 2 √ 2 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • D ⎛ ⎜ ⎜ ⎜ ⎝ − √ 2 2 − √ 2 2 − √ 2 2 √ 2 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • E ⎛ ⎜ ⎜ ⎜ ⎝ − √ 2 2 0 0 √ 2 2 ⎞ ⎟ ⎟ ⎟ ⎠

Q3:

A linear transformation is formed by rotating every vector in ℝ 2 through an angle of 𝜋 6 , reflecting the resulting vector in the 𝑥 -axis, and finally reflecting this vector in the 𝑦 -axis. Find the matrix of this linear transformation.

  • A ⎛ ⎜ ⎜ ⎜ ⎝ √ 3 2 − 1 2 − 1 2 − √ 3 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • B ⎛ ⎜ ⎜ ⎜ ⎝ − √ 3 2 − 1 2 1 2 √ 3 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • C ⎛ ⎜ ⎜ ⎜ ⎝ √ 3 2 − 1 2 1 2 √ 3 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • D ⎛ ⎜ ⎜ ⎜ ⎝ − √ 3 2 1 2 − 1 2 − √ 3 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • E ⎛ ⎜ ⎜ ⎜ ⎝ − √ 3 2 0 0 √ 3 2 ⎞ ⎟ ⎟ ⎟ ⎠

Q4:

A linear transformation is formed by rotating every vector in ℝ 2 through an angle of 2 𝜋 3 and then reflecting the resulting vector in the 𝑥 -axis. Find the matrix of this linear transformation.

  • A ⎛ ⎜ ⎜ ⎜ ⎝ − 1 2 − √ 3 2 √ 3 2 − 1 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • B ⎛ ⎜ ⎜ ⎜ ⎝ − 1 2 √ 3 2 − √ 3 2 1 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • C ⎛ ⎜ ⎜ ⎝ − 1 2 0 0 1 2 ⎞ ⎟ ⎟ ⎠
  • D ⎛ ⎜ ⎜ ⎜ ⎝ − 1 2 − √ 3 2 − √ 3 2 1 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • E ⎛ ⎜ ⎜ ⎜ ⎝ 1 2 − √ 3 2 − √ 3 2 − 1 2 ⎞ ⎟ ⎟ ⎟ ⎠

Q5:

A linear transformation is formed by reflecting every vector in ℝ 2 across the 𝑦 -axis and then rotating the resulting vector through an angle of 𝜋 6 . Find the matrix of this linear transformation.

  • A ⎛ ⎜ ⎜ ⎜ ⎝ − √ 3 2 1 2 1 2 √ 3 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • B ⎛ ⎜ ⎜ ⎜ ⎝ − √ 3 2 1 2 − 1 2 √ 3 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • C ⎛ ⎜ ⎜ ⎜ ⎝ √ 3 2 − 1 2 1 2 √ 3 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • D ⎛ ⎜ ⎜ ⎜ ⎝ − √ 3 2 − 1 2 − 1 2 √ 3 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • E ⎛ ⎜ ⎜ ⎜ ⎝ − √ 3 2 0 0 √ 3 2 ⎞ ⎟ ⎟ ⎟ ⎠

Q6:

A linear transformation is formed by rotating every vector in ℝ 2 through an angle of 𝜋 3 and then reflecting the resulting vector in the 𝑥 -axis. Find the matrix of this linear transformation.

  • A ⎛ ⎜ ⎜ ⎜ ⎝ 1 2 − √ 3 2 √ 3 2 1 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • B ⎛ ⎜ ⎜ ⎜ ⎝ 1 2 √ 3 2 − √ 3 2 − 1 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • C ⎛ ⎜ ⎜ ⎝ 1 2 0 0 − 1 2 ⎞ ⎟ ⎟ ⎠
  • D ⎛ ⎜ ⎜ ⎜ ⎝ 1 2 − √ 3 2 − √ 3 2 − 1 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • E ⎛ ⎜ ⎜ ⎜ ⎝ − 1 2 − √ 3 2 − √ 3 2 1 2 ⎞ ⎟ ⎟ ⎟ ⎠

Q7:

A linear transformation is formed by reflecting every vector in ℝ 2 in the 𝑥 -axis and then rotating the resulting vector through an angle of 𝜋 6 . Find the matrix of this linear transformation.

  • A ⎛ ⎜ ⎜ ⎜ ⎝ √ 3 2 − 1 2 − 1 2 − √ 3 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • B ⎛ ⎜ ⎜ ⎜ ⎝ √ 3 2 − 1 2 1 2 − √ 3 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • C ⎛ ⎜ ⎜ ⎜ ⎝ √ 3 2 − 1 2 1 2 √ 3 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • D ⎛ ⎜ ⎜ ⎜ ⎝ √ 3 2 1 2 1 2 − √ 3 2 ⎞ ⎟ ⎟ ⎟ ⎠
  • E ⎛ ⎜ ⎜ ⎜ ⎝ √ 3 2 0 0 − √ 3 2 ⎞ ⎟ ⎟ ⎟ ⎠

Q8:

A linear transformation is formed by reflecting every vector in ℝ in the 𝑥 -axis and then rotating the resulting vector through an angle of 𝜋 4 . Find the matrix of this linear transformation.

  • A ⎡ ⎢ ⎢ ⎢ ⎣ √ 2 2 − √ 2 2 − √ 2 2 − √ 2 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • B ⎡ ⎢ ⎢ ⎢ ⎣ √ 2 2 − √ 2 2 √ 2 2 − √ 2 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • C ⎡ ⎢ ⎢ ⎢ ⎣ √ 2 2 − √ 2 2 √ 2 2 √ 2 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • D ⎡ ⎢ ⎢ ⎢ ⎣ √ 2 2 √ 2 2 √ 2 2 − √ 2 2 ⎤ ⎥ ⎥ ⎥ ⎦
  • E ⎡ ⎢ ⎢ ⎢ ⎣ √ 2 2 0 0 − √ 2 2 ⎤ ⎥ ⎥ ⎥ ⎦

Q9:

Let the matrix represent rotation in the plane through an angle of and let the matrix represent reflection in the -axis.

What is the matrix ?

  • A
  • B
  • C
  • D
  • E

What is the matrix ?

  • A
  • B
  • C
  • D
  • E

What is the matrix ?

  • A
  • B
  • C
  • D
  • E

Q10:

Suppose that the matrix represents rotation about the origin through an angle of (measuring between and ) and the matrix represents reflection in the -axis.

Find .

  • A
  • B
  • C
  • D
  • E

Note that is a reflection in a line through the origin. Let this line of reflection have equation . By considering the image of the vector , determine the measure of the angle between the -axis and the line of reflection.

  • A
  • B
  • C
  • D
  • E

What, therefore, is the slope of the line of reflection?

  • A
  • B
  • C
  • D
  • E

If lies in the direction of the line of reflection, what is ?

  • A
  • B
  • C
  • D
  • E

By solving the equation obtained in the previous part, find another expression for the slope of the line of reflection.

  • A
  • B
  • C
  • D
  • E

Q11:

Consider the given figure.

The points , , , and are corners of the unit square. This square is reflected in the line with equation to form the image .

As is the image of in the line through and , . Use this fact and the identity to find the gradient and hence equation of from the gradient of .

  • A
  • B
  • C
  • D
  • E

Using the fact that is perpendicular to , find the equation of .

  • A
  • B
  • C
  • D
  • E

Using the fact that , find the coordinates of and .

  • A ,
  • B ,
  • C ,
  • D ,
  • E ,

Using the fact that a reflection in a line through the origin is a linear transformation, find the matrix which represents reflection in the line .

  • A
  • B
  • C
  • D
  • E

Q12:

A linear transformation is formed by rotating every vector in ℝ 3 through an angle of 3 0 ∘ counterclockwise (when viewed from the positive 𝑧 -axis) about the 𝑧 -axis and then reflecting the resulting vector in the 𝑥 𝑦 -plane. Find the matrix of this linear transformation.

  • A ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ √ 3 2 − 1 2 0 1 2 √ 3 2 0 0 0 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠
  • B ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 1 2 − √ 3 2 0 √ 3 2 1 2 0 0 0 − 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠
  • C ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 1 2 − √ 3 2 0 √ 3 2 1 2 0 0 0 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠
  • D ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ √ 3 2 − 1 2 0 1 2 √ 3 2 0 0 0 − 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠
  • E ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ √ 3 2 − 1 2 0 − 1 2 − √ 3 2 0 0 0 − 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠

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